Linearly Coupled Kirchhoff–Choquard System

• The paper establishes the existence of positive ground state solutions for the Kirchhoff–Choquard system in subcritical and half-critical regimes using a Nehari–Pohozaev manifold.
• It introduces a novel variational framework that blends Kirchhoff-type global coefficients with Choquard convolution nonlinearities via a unique fiber map scaling to handle compactness challenges.
• The work rigorously identifies parameter thresholds and coupling conditions essential for recovering compactness, while also clarifying nonexistence at simultaneous critical endpoints.

Searching arXiv for the specified paper and closely related Kirchhoff–Choquard work. arXiv.search_^^^^^^^^6query6 of ground state solutions to Kirchhoff--Choquard system in R^^^^^6max_results6^^^^ with constant potentials\"^^^^^^^^6 OR id:(Matsuzawa, 12 Jul 2025)6^^^^^^^^"," arXiv.search_^^^^^^^^6query6 Choquard system\"","^^^^^^^^6max_results6 arXiv.search_^^^^^^^^6query6 Choquard\" AND (^^^^^^^^6all:\6 Pohozaev\" OR ^^^^^^^^6all:\6 Liu\")","^^^^^^^^6max_results6 The linearly coupled Kirchhoff--Choquard system is a nonlinear, nonlocal elliptic system in which Kirchhoff-type global coefficients and Choquard-type convolution nonlinearities are combined with a linear intercomponent coupling. In the constant-potential setting studied in PRESERVED_PLACEHOLDER_^^^^6query6^^^^, the model takes the form (&&&^^^^6query6^^^^&&&)

PRESERVED_PLACEHOLDER_^^^^6ti:\6^^^^

where PRESERVED_PLACEHOLDER_^^^^6 OR id:(Matsuzawa, 12 Jul 2025)6^^^^ and PRESERVED_PLACEHOLDER_^^^^6max_results6^^^^. The analysis in (&&&^^^^6query6^^^^&&&) is centered on the existence of positive ground state solutions, the role of the Nehari--Pohozaev manifold as a natural variational constraint, the recovery of compactness in half-critical regimes through large-parameter conditions, and nonexistence at the simultaneous critical endpoints.

^^^^6ti:\6^^^^. Structural form of the system

The system is simultaneously nonlocal in two distinct senses. The Kirchhoff contribution enters through the coefficients

PRESERVED_PLACEHOLDER_^^^^6query6^^^^

which depend on the global Dirichlet energy of the unknown; this is the source of the Kirchhoff character. The Choquard contribution enters through the Riesz-potential convolution

PRESERVED_PLACEHOLDER_^^^^6all:\6^^^^

which generates the Hartree-type nonlocal interaction. The linear terms $\lambda v$ and $\lambda u$ provide the intercomponent coupling (&&&^^^^6query6^^^^&&&).

In $\mathbb{R}^3$, the Riesz kernel is

$$I_\alpha(x)=A_{3,\alpha}|x|^{\alpha-3}, \qquad A_{3,\alpha}= \frac{\Gamma((3-\alpha)/2)}{\Gamma(\alpha/2)\pi^{3/2}2^\alpha}.$$

Accordingly, the Choquard energy can also be written in symmetric double-integral form; for example,

PRESERVED_PLACEHOLDER_^^^^6ti:\6query6^^^^

The exponent structure is decisive. The treatment in (&&&^^^^6query6^^^^&&&) covers three regimes: PRESERVED_PLACEHOLDER_^^^^6ti:\6ti:\6^^^^ The first is the strictly subcritical regime, the second the upper half critical regime, and the third the lower half critical regime. By contrast, the simultaneous endpoints PRESERVED_PLACEHOLDER_^^^^6ti:\6 OR id:(Matsuzawa, 12 Jul 2025)6^^^^ and PRESERVED_PLACEHOLDER_^^^^6ti:\6max_results6^^^^ are shown to admit no nontrivial solution, while the mixed endpoint PRESERVED_PLACEHOLDER_^^^^6ti:\6query6^^^^ remains open (&&&^^^^6query6^^^^&&&).

^^^^6 OR id:(Matsuzawa, 12 Jul 2025)6^^^^. Variational framework and notions of solution

The natural energy space is

PRESERVED_PLACEHOLDER_^^^^6ti:\6all:\6^^^^

equipped with the norm

PRESERVED_PLACEHOLDER_^^^^6ti:\66^^^^

where

PRESERVED_PLACEHOLDER_^^^^6ti:\6^^^^7

This is an equivalent norm on PRESERVED_PLACEHOLDER_^^^^6ti:\6^^^^8 for each component (&&&^^^^6query6^^^^&&&).

A weak solution is a pair PRESERVED_PLACEHOLDER_^^^^6ti:\6^^^^9 that is a critical point of the associated energy functional PRESERVED_PLACEHOLDER_^^^^6 OR id:(Matsuzawa, 12 Jul 2025)6query6^^^^, meaning that

PRESERVED_PLACEHOLDER_^^^^6 OR id:(Matsuzawa, 12 Jul 2025)6ti:\6^^^^

A nontrivial weak solution PRESERVED_PLACEHOLDER_^^^^6 OR id:(Matsuzawa, 12 Jul 2025)6 OR id:(Matsuzawa, 12 Jul 2025)6^^^^ is a ground state if

PRESERVED_PLACEHOLDER_^^^^6 OR id:(Matsuzawa, 12 Jul 2025)6max_results6^^^^

for every other nontrivial weak solution PRESERVED_PLACEHOLDER_^^^^6 OR id:(Matsuzawa, 12 Jul 2025)6query6^^^^ (&&&^^^^6query6^^^^&&&).

The energy functional is

PRESERVED_PLACEHOLDER_^^^^6 OR id:(Matsuzawa, 12 Jul 2025)6all:\6^^^^

Using the Hardy--Littlewood--Sobolev inequality together with the Sobolev embeddings PRESERVED_PLACEHOLDER_^^^^6 OR id:(Matsuzawa, 12 Jul 2025)66^^^^ for PRESERVED_PLACEHOLDER_^^^^6 OR id:(Matsuzawa, 12 Jul 2025)6^^^^7, (&&&^^^^6query6^^^^&&&) shows that PRESERVED_PLACEHOLDER_^^^^6 OR id:(Matsuzawa, 12 Jul 2025)6^^^^8 is finite and PRESERVED_PLACEHOLDER_^^^^6 OR id:(Matsuzawa, 12 Jul 2025)6^^^^9 on PRESERVED_PLACEHOLDER_^^^^6max_results6query6^^^^ when

PRESERVED_PLACEHOLDER_^^^^6max_results6ti:\6^^^^

A structural assumption on the linear coupling is imposed throughout: PRESERVED_PLACEHOLDER_^^^^6max_results6 OR id:(Matsuzawa, 12 Jul 2025)6^^^^ Under this condition,

PRESERVED_PLACEHOLDER_^^^^6max_results6max_results6^^^^

so the coupling is controlled by the positive potential terms. This coercivity mechanism is fundamental for the lower bounds on the variational constraint set and for positivity of the minimization level (&&&^^^^6query6^^^^&&&).

^^^^6max_results6^^^^. Nehari--Pohozaev manifold and scaling geometry

The variational analysis is organized around a scaling tailored to the Kirchhoff structure: PRESERVED_PLACEHOLDER_^^^^6max_results6query6^^^^ For a fixed nonzero pair PRESERVED_PLACEHOLDER_^^^^6max_results6all:\6^^^^, one defines the fiber map

PRESERVED_PLACEHOLDER_^^^^6max_results66^^^^

Direct computation yields

PRESERVED_PLACEHOLDER_^^^^6max_results6^^^^7

For PRESERVED_PLACEHOLDER_^^^^6max_results6^^^^8, the fiber map has a unique critical point PRESERVED_PLACEHOLDER_^^^^6max_results6^^^^9, and that point is the global maximum on PRESERVED_PLACEHOLDER_^^^^6query6query6^^^^ (&&&^^^^6query6^^^^&&&).

The Pohozaev identity plays a central role. Every weak solution satisfies

PRESERVED_PLACEHOLDER_^^^^6query6ti:\6^^^^

Differentiating PRESERVED_PLACEHOLDER_^^^^6query6 OR id:(Matsuzawa, 12 Jul 2025)6^^^^ at PRESERVED_PLACEHOLDER_^^^^6query6max_results6^^^^ produces the Nehari--Pohozaev functional

PRESERVED_PLACEHOLDER_^^^^6query6query6^^^^

It satisfies

PRESERVED_PLACEHOLDER_^^^^6query6all:\6^^^^

The associated constraint is the Nehari--Pohozaev manifold

PRESERVED_PLACEHOLDER_^^^^6query66^^^^

Every nontrivial critical point of PRESERVED_PLACEHOLDER_^^^^6query6^^^^7 belongs to PRESERVED_PLACEHOLDER_^^^^6query6^^^^8, and for every PRESERVED_PLACEHOLDER_^^^^6query6^^^^9 there exists a unique PRESERVED_PLACEHOLDER_^^^^6all:\6query6^^^^ such that PRESERVED_PLACEHOLDER_^^^^6all:\6ti:\6^^^^. On PRESERVED_PLACEHOLDER_^^^^6all:\6 OR id:(Matsuzawa, 12 Jul 2025)6^^^^,

PRESERVED_PLACEHOLDER_^^^^6all:\6max_results6^^^^

This converts the search for ground states into a constrained minimization problem on a set adapted to both the Euler--Lagrange equation and the Pohozaev identity (&&&^^^^6query6^^^^&&&).

Under the coupling condition PRESERVED_PLACEHOLDER_^^^^6all:\6query6^^^^, the manifold is separated from the origin: there exists PRESERVED_PLACEHOLDER_^^^^6all:\6all:\6^^^^ such that

PRESERVED_PLACEHOLDER_^^^^6all:\66^^^^

Moreover, for PRESERVED_PLACEHOLDER_^^^^6all:\6^^^^7,

PRESERVED_PLACEHOLDER_^^^^6all:\6^^^^8

and hence

PRESERVED_PLACEHOLDER_^^^^6all:\6^^^^9

^^^^6query6^^^^. Existence, half-critical thresholds, and endpoint nonexistence

The ground-state problem is formulated as

$\lambda v$^^^^6query6^^^^

Within this framework, (&&&^^^^6query6^^^^&&&) establishes the following existence and nonexistence picture.

Exponent regime	Assumptions	Conclusion
$\lambda v$^^^^6ti:\6^^^^	$\lambda v$^^^^6 OR id:(Matsuzawa, 12 Jul 2025)6^^^^, any $\lambda v$^^^^6max_results6^^^^	positive ground state solution
$\lambda v$^^^^6query6^^^^	$\lambda v$^^^^6all:\6^^^^, fixed $\lambda v$6, $\lambda v$7	ground state solution
$\lambda v$8	$\lambda v$9, fixed $\lambda u$^^^^6query6^^^^, $\lambda u$^^^^6ti:\6^^^^	ground state solution
$\lambda u$^^^^6 OR id:(Matsuzawa, 12 Jul 2025)6^^^^ or $\lambda u$^^^^6max_results6^^^^	$\lambda u$^^^^6query6^^^^	no nontrivial solution
$\lambda u$^^^^6all:\6^^^^	—	open

In the upper half critical case $\lambda u$6, compactness is recovered by forcing the minimization level below the sharp threshold associated with the best constant $\lambda u$7, defined by

$\lambda u$8

For each fixed $\lambda u$9, there exists $\mathbb{R}^3$^^^^6query6^^^^ such that, for $\mathbb{R}^3$^^^^6ti:\6^^^^,

$\mathbb{R}^3$^^^^6 OR id:(Matsuzawa, 12 Jul 2025)6^^^^

This strict inequality is the quantitative device ensuring that concentration cannot be lost at the critical level (&&&^^^^6query6^^^^&&&).

In the lower half critical case $\mathbb{R}^3$^^^^6max_results6^^^^, the relevant sharp constant is

$\mathbb{R}^3$^^^^6query6^^^^

with

$\mathbb{R}^3$^^^^6all:\6^^^^

For each fixed $\mathbb{R}^3$6, there exists $\mathbb{R}^3$7 such that, for $\mathbb{R}^3$8,

$\mathbb{R}^3$9

The role of these large-parameter assumptions is not merely technical: they ensure that the ground-state level lies below the critical compactness threshold. This suggests that the subcritical component can be used to dominate the variational level even when one Choquard exponent reaches a critical boundary.

^^^^6all:\6^^^^. Compactness, splitting, regularity, and positivity

The existence proof proceeds by minimizing $$I_\alpha(x)=A_{3,\alpha}|x|^{\alpha-3}, \qquad A_{3,\alpha}= \frac{\Gamma((3-\alpha)/2)}{\Gamma(\alpha/2)\pi^{3/2}2^\alpha}.$$^^^^6query6^^^^ on $$I_\alpha(x)=A_{3,\alpha}|x|^{\alpha-3}, \qquad A_{3,\alpha}= \frac{\Gamma((3-\alpha)/2)}{\Gamma(\alpha/2)\pi^{3/2}2^\alpha}.$$^^^^6ti:\6^^^^. Starting from a minimizing sequence $$I_\alpha(x)=A_{3,\alpha}|x|^{\alpha-3}, \qquad A_{3,\alpha}= \frac{\Gamma((3-\alpha)/2)}{\Gamma(\alpha/2)\pi^{3/2}2^\alpha}.$$^^^^6 OR id:(Matsuzawa, 12 Jul 2025)6^^^^ with $$I_\alpha(x)=A_{3,\alpha}|x|^{\alpha-3}, \qquad A_{3,\alpha}= \frac{\Gamma((3-\alpha)/2)}{\Gamma(\alpha/2)\pi^{3/2}2^\alpha}.$$^^^^6max_results6^^^^, the argument combines concentration--compactness in the sense of Lions with a nonlocal Brezis--Lieb splitting for the Choquard terms (&&&^^^^6query6^^^^&&&). The splitting formula has the form

$$I_\alpha(x)=A_{3,\alpha}|x|^{\alpha-3}, \qquad A_{3,\alpha}= \frac{\Gamma((3-\alpha)/2)}{\Gamma(\alpha/2)\pi^{3/2}2^\alpha}.$$^^^^6query6^^^^

together with analogous decompositions for $$I_\alpha(x)=A_{3,\alpha}|x|^{\alpha-3}, \qquad A_{3,\alpha}= \frac{\Gamma((3-\alpha)/2)}{\Gamma(\alpha/2)\pi^{3/2}2^\alpha}.$$^^^^6all:\6^^^^ and $$I_\alpha(x)=A_{3,\alpha}|x|^{\alpha-3}, \qquad A_{3,\alpha}= \frac{\Gamma((3-\alpha)/2)}{\Gamma(\alpha/2)\pi^{3/2}2^\alpha}.$$6. Because of the Kirchhoff terms, these identities contain mixed quadratic--quartic gradient contributions that do not appear in non-Kirchhoff Choquard systems.

Nonvanishing is obtained by contradiction. If local mass vanished, then the Hardy--Littlewood--Sobolev estimate would force the Choquard energies to vanish, contradicting the positive lower bound on $$I_\alpha(x)=A_{3,\alpha}|x|^{\alpha-3}, \qquad A_{3,\alpha}= \frac{\Gamma((3-\alpha)/2)}{\Gamma(\alpha/2)\pi^{3/2}2^\alpha}.$$7. In the half-critical regimes, the contradiction requires the sharp constants $$I_\alpha(x)=A_{3,\alpha}|x|^{\alpha-3}, \qquad A_{3,\alpha}= \frac{\Gamma((3-\alpha)/2)}{\Gamma(\alpha/2)\pi^{3/2}2^\alpha}.$$8 or $$I_\alpha(x)=A_{3,\alpha}|x|^{\alpha-3}, \qquad A_{3,\alpha}= \frac{\Gamma((3-\alpha)/2)}{\Gamma(\alpha/2)\pi^{3/2}2^\alpha}.$$9 together with the threshold inequalities above. Once a nontrivial weak limit is recovered, a deformation argument and degree theory show that a minimizer on PRESERVED_PLACEHOLDER_^^^^6ti:\6query6query6^^^^ is in fact a free critical point of PRESERVED_PLACEHOLDER_^^^^6ti:\6query6ti:\6^^^^ in PRESERVED_PLACEHOLDER_^^^^6ti:\6query6 OR id:(Matsuzawa, 12 Jul 2025)6^^^^, so the constrained minimizer is a genuine weak solution (&&&^^^^6query6^^^^&&&).

A notable technical contribution is the regularity theorem for coupled Choquard systems. For PRESERVED_PLACEHOLDER_^^^^6ti:\6query6max_results6^^^^, PRESERVED_PLACEHOLDER_^^^^6ti:\6query6query6^^^^, if PRESERVED_PLACEHOLDER_^^^^6ti:\6query6all:\6^^^^ solves

PRESERVED_PLACEHOLDER_^^^^6ti:\6query66^^^^

then

PRESERVED_PLACEHOLDER_^^^^6ti:\6query6^^^^7

This regularity justifies the localized virial-type testing needed for the Pohozaev identity. The proof adapts Moroz--Van Schaftingen bootstrap arguments, a coupled linear estimate, and a Choquard-specific Gagliardo--Nirenberg--HLS interpolation cited in (&&&^^^^6query6^^^^&&&).

Positivity of ground states follows from a variational and PDE argument. From a ground state PRESERVED_PLACEHOLDER_^^^^6ti:\6query6^^^^8, one may take absolute values and rescale along the fiber to obtain PRESERVED_PLACEHOLDER_^^^^6ti:\6query6^^^^9 such that PRESERVED_PLACEHOLDER_^^^^6ti:\6ti:\6query6^^^^ and

PRESERVED_PLACEHOLDER_^^^^6ti:\6ti:\6ti:\6^^^^

Then the strong maximum principle, applied componentwise in the cooperative setting, yields strict positivity.

6. Relation to adjacent theories and unresolved directions

The analysis in (&&&^^^^6query6^^^^&&&) is explicitly positioned at the intersection of two earlier variational developments. Ueno, in Communications on Pure and Applied Analysis ^^^^6 OR id:(Matsuzawa, 12 Jul 2025)6query6^^^^ (^^^^6 OR id:(Matsuzawa, 12 Jul 2025)6query6 OR id:(Matsuzawa, 12 Jul 2025)6all:\6^^^^), established ground states for linearly coupled Kirchhoff--Schrödinger systems with local nonlinearities by means of the Nehari--Pohozaev manifold, including critical exponents PRESERVED_PLACEHOLDER_^^^^6ti:\6ti:\6 OR id:(Matsuzawa, 12 Jul 2025)6^^^^ with large PRESERVED_PLACEHOLDER_^^^^6ti:\6ti:\6max_results6^^^^. Chen--Liu, in Journal of Mathematical Analysis and Applications ^^^^6query6max_results6max_results6^^^^ (^^^^6 OR id:(Matsuzawa, 12 Jul 2025)6query6ti:\6all:\12^^^^, developed the Nehari--Pohozaev method for single Kirchhoff--Choquard equations with

PRESERVED_PLACEHOLDER_^^^^6ti:\6ti:\6query6^^^^

The linearly coupled Kirchhoff--Choquard system combines these two directions: it incorporates Kirchhoff nonlocality, Choquard nonlocality, and linear coupling in a single framework, while extending the Nehari--Pohozaev machinery to the coupled nonlocal setting (&&&^^^^6query6^^^^&&&).

Relative to standard non-Kirchhoff Choquard systems, the Kirchhoff terms alter both scaling and compactness. The fiber map contains PRESERVED_PLACEHOLDER_^^^^6ti:\6ti:\6all:\6^^^^-terms arising from the quartic gradient contributions, and the energy splitting must accommodate the interaction between weak limits and the global coefficients

PRESERVED_PLACEHOLDER_^^^^6ti:\6ti:\66^^^^

This changes the variational landscape and the critical threshold analysis. A plausible implication is that even familiar Choquard concentration phenomena must be re-evaluated once Kirchhoff modulation is present, because the natural scaling and the compactness defects are no longer the same as in the purely semilinear Hartree setting.

Several limitations remain explicit. The potentials PRESERVED_PLACEHOLDER_^^^^6ti:\6ti:\6^^^^7 are assumed to be positive constants. Variable potentials are reserved for forthcoming work, where the present concentration--compactness argument would require refinement because additional localization issues arise. Radial symmetry, uniqueness, multiplicity, decay at infinity, and nondegeneracy are not treated. The simultaneous critical endpoints admit only the trivial solution, whereas the mixed endpoint

PRESERVED_PLACEHOLDER_^^^^6ti:\6ti:\6^^^^8

remains open (&&&^^^^6query6^^^^&&&).

Within the current constant-potential theory, the main conclusion is precise: under

PRESERVED_PLACEHOLDER_^^^^6ti:\6ti:\6^^^^9

the system admits positive ground state solutions throughout the subcritical regime and in the two half-critical regimes under sufficiently large PRESERVED_PLACEHOLDER_^^^^6ti:\6 OR id:(Matsuzawa, 12 Jul 2025)6query6^^^^ or PRESERVED_PLACEHOLDER_^^^^6ti:\6 OR id:(Matsuzawa, 12 Jul 2025)6ti:\6^^^^, while nontrivial solutions fail at the simultaneous critical endpoints. The combination of the Nehari--Pohozaev manifold, concentration--compactness, sharp HLS-type thresholds, and coupled Choquard regularity constitutes the core framework for this result (&&&^^^^6query6^^^^&&&).